Let , and denote by the second intersection of and , where denotes the circumcircle of triangle .
We know that is the center of a spiral similarity taking to - therefore, using directed angles , we find that .
Since , reciprocating and adding to both sides yields , so . The spiral similarity taking to from gives , so . Combining this with , we find that is the center of a spiral similarity taking to , so is also the center of a spiral similarity taking to . Thus , so also lies on . By Miquel's Theorem on quadrilaterals and , we find that is also the second intersection point of and , and the second intersection point of and , so lies on and , completing the proof.

So yesterday, I spent pretty much the entire day in a water polo tournament. We won the first game by a landslide... and we lost the second game by quite a bit. Now, it's the third game, and we're playing for third place and a nice big trophy. Just our luck, we're playing our coach's friend's team.
First quarter: We're down 0-3. Not looking too good for us.
Second quarter: Still 0-3. 0-0 quarter. We've still got some fighting to do.
Third quarter: We tied it up 3-3, thanks to two amazing goals from probably the best guy on our team in the very beginning of the quarter.
Fourth quarter: Here's where it gets intense. 5-5. Yes, it's tied, after the fourth quarter.
Overtime: Other team wins the sprint. I guard set position. Ball gets into set. Set backhands it, but an amazing block from our goalie saves us the game. We sprint down the pool, and I receive a pass. I'm at the 2 spot, where I have basically the second best shooting angle in the entire field (except for set). My defender drops to cover set, and I'm left with a wide open shot. I'm six meters from the cage. The defender is coming out. I fake once. Twice. Three times. Here it comes. I get up, and I shoot the ball. The goalie tips it, and like it's in slow motion, the ball slips from his hands into the net. The ball is scored, and since it's sudden death, that's the game, and that's our trophy.
What a game. Afterwards, the goalie and I were nominated as the Tournament MVPs for our team (considering the first and second games as well), and it was just a great day. Truly a team effort - our defense in the second quarter pulled us through and gave us some hope. During the third quarter, the two goals from A right away got us pushing, and we tied it up. The fourth quarter is where we tied it up, and it was overtime where our collective defense and our goalie saved us the game.